fo(z). fo(z) fz (0, t)>0. f(O,t) f(z,O) = fo(z) Re(1-~t(z,t)-~z(Z,t))=l f(z,t), fo(z), A simplified proof for a moving boundary problem for Hele-Shaw flows in the plane

A simplified proof for a moving boundary problem for Hele-Shaw flows in the plane

Michael Reissig and Lothar v. Wolfersdorf

0. I n t r o d u c t i o n

In [7] Richardson derived a mathematical model for describing Hele-Shaw flows with a free b o u n d a r y produced by the injection of fluid into a narrow channel.

This model can be represented in the following form (see also [3]): Given

fo(z),

f 0 ( 0 ) = 0 , analytic and univalent in a neighbourhood of Izl_<l, find

f(z,t),

ana- lytic and univalent as a function of z in a neighbourhood of Izl_<l, continuously differentiable with respect to t in a right-sided neighbourhood of t = 0 , satisfying

(1)

Re(1-~t(z,t)-~z(Z,t))=l

for N - - 1 ;

(2)

f(z,O) = fo(z)

for Izl

_< 1;

(3)

f(O,t)

= 0 .

With the results of V i n o g r a d o ~ K u f a r e v [9] one gets the existence and unique- ness of solutions which depend analytically on z and t under the additional assump- tion

fz (0, t)>0.

But the proofs in [9] are fairly complicated.

For this reason Gustafsson gave in [3] a more elementary proof of existence and uniqueness of solutions of (1)-(3) in the case that

fo(z)

is a polynomial or a rational function. In b o t h cases the solution is of the same sort with regard to z as the initial value

fo(z).

T h e restriction to rational initial values seems to be indispensable for the used reduction of (1) to a finite system of ordinary differential equations in t.

T h e goal of the present paper is to give a simplified proof for a generalized Hele-Shaw problem containing as a special case the above formulated problem (1)- (3). This proof is based on the application of the non-linear abstract C a u c h y - Kovalevsky theorem which was proved by Nishida in [5]. Moreover, this theorem gives uniqueness for solutions depending continuously differentiably on t.

102 Michael Reissig and Lothar v. Wolfersdorf

T h e o r e m 1 ([5]). Let us consider the abstract Cauchy-Kovalevsky problem dw w), w(O) = 0

(4) =g(t,

satisfying the following conditions in a scale of Banach spaces { B~,

II'll~}o<~_<l

(A family of continuously embedded Banach spaces {B~, H'l[s}0<~_<l is called a Ba- nach space scale if for all 0 < s ~ < s < 1 the norm of the canonical embedding operator IlI~--*s ' II ~- 1.) ( C, K, R and T are certain positive constants independent of s', s, t ) :

(i) the right-hand side s w) is a continuous, in t, mapping of (5) [ O , T ] • ) intoB~, f o r a l l O < s ' < s < l ;

(ii) the continuous function s O) satisfies

(6) Ilk(t, 0)lls <_ K / ( 1 - s ) for all 0 < s < 1;

(iii) for all 0 < s ' < s < l , t 9 and Wl,W2 belonging to {Hwlls<R} we have

(7)

Under these assumptions there exists one and only one solution

W 9 c l ( [ 0 , a 0 ( 1 - s ) ) ,

Bs)o<s<l,

Ilw(t)lls <

R,

where ao is a suitable positive constant.

This theorem represents an essential tool for solving non-linear time-dependent mixed problems for harmonic or holomorphic functions in the mathematical litera- ture ([1, 2, 4, 6]). Our problem (1)-(3) is of such a type. We shall show that after the reduction of the generalized Hele-Shaw problem to an equivalent problem for w = ( O f / O z ) -1, which fulfills all the conditions (5)-(7) in suitable scales of Banach spaces, the abstract theorem is applicable and yields immediately the main result of [9] as a special case.

The result of Gustafsson [3] can be interpreted as a regularity result concerning the corresponding structures of the initial value and the solution. A result of the same type is derived at the end of this paper for (Of/Oz) -~ or (Ofo/Oz) -1 belonging to special classes of entire functions.

A simplified proof for a moving boundary problem for Hele-Shaw flows in the plane 103 1. H e u r i s t i c c o n s i d e r a t i o n s a n d t h e d e r i v a t i o n o f a s c a l e - t y p e p r o b l e m

Let us start with a generalization of (1) to

(8)

Re(h(lt) ~t (z ' Of t)~z (Z,t) ) =g(z,~,t)

for all ]zI----1 and t > 0 , where

(i) the real-valued function

g=g(z,2, t)

is continuous on {Izl=l} x [0, T] and possesses a holomorphic extension from IzI--1 into a circular ring

(9)

Kb={1/b<lz]<b},

b > l , for all t E [0, T];

(ii) the function

h=h(z, t)

is continuous in tE [0, T] and for each such t analytic in a neighbourhood of

(10) IzI_<l, h(0, t ) = 0 ,

hz(O,t)~O

for all t E [0, T]

and

h(z,t)7~O

for a l l ( z , t ) E { 0 < i z i < _ l } x [ 0 , T

].

Setting

h(z, t)=z

^and

g(z, 2,

t)--1 in (8) we have the condition (1). The con- dition (8) is equivalent to

Of (z,t)(~z)-l(z,t)) Of(z,

From the assumptions (3), (9), (10) and the univalence of

f(z, t)

in a neighbourhood of {Izl_<l} for all t~[0, T] we get the holomorphy of

Of (z, t) ( ~ Yl(z, t)/h(z, t)

ot \ Oz ]

in {]z]<l}. Using (8) and the fact that every holomorphic function in { ] z l < l } with prescribed real part on {]z]=l} is uniquely determined by the value for the imaginary part in z--0 we are able to formulate the additional condition

Im( h(lt) O~f~ ~, v~ 0 -1

(11)

The application of the Schwarz formula leads to

(12)

Of O-[(z't)-h(z't)-~z (Z't)2-~i __1=1 Og Of 1 /z Of -2g(L), ^{~,t) p+z dg} Q-z 0

for Izl < 1. For our further investigations we need the space 7-/(Gr)MC(Gr), that is the space of all complex-valued functions defined and continuous in G~ and holomor- phic in G r = { i z I <r}. In the same manner we introduce the spaces

7-I(Gr)NC~(G~),

7-/(G~)MCI(G~) and 7-/(G~)nCI,~(G~).

104 Michael Reissig and Lothar v. Wolfersdorf

L e m m a 1.

Let us suppose that f(z, t)

9

ao),7-l(Cl)ACZ(Gz)) is for each

t 9 [0,

ao) a univalent function in

Iz[_< 1

and in

Gz x (0,

ao) a solution of the problem

(8), (11), (2)

and

(3),

and equivalently, of the problem

(12), (2)

and

(3).

Then v(z, t)=(Of /Oz) -1

9 ao),

7-I(G1)NC(G1)) is a solution of

(13)

Ov Ov 0

Ot hTt(V)Oz+V-~z(hTt(v))=O for (z,t)eal • ao),

(14)

v(z,O)

=v0(z) =

(Ofo/Oz) -1 for z 9 G1, where v(z,

t ) r

Here Tt(v) denotes the non-linear operator

s

1 ]v(P)]2g(P, O, t)

(15) r (v) := al

m

Conversely, let us suppose that v(z,t)eCl([O, al),Tl(G1)nC(Gz)) is a solu- tion of

(13)

and

(14)

with v(z,t)r in

G l x [ 0 , ao).

Then f(z,t)=fo(dP)/(v(p,t)) belonging to Cl ([O, ao ), ~-~( G1)NCl ( a l ) ) represents a locally univalent solution of

(12), (2),

and

(3)

and, equivalently, of

(8), (11), (2)

and

(3)

in

GlX [0, a0).

Proof.

Let

f = f ( z , t)

as a univalent solution of (12), (2) and (3) satisfy the con- ditions of this lemma. Then

v=(Of/Oz) -1

belongs to

Cl([O, ao),7-l(G1)NC(G1)).

Differentiating (12) with respect to z, one obtains with v =

(Of/Oz) -~

O(1/v) hTt(v) O(~zV ) 1 0 ( h T t ( v ) ) = O '

Ot v

and hence,

Ov hTt(V)~zz+VO (hTt(v))=O with v(z,O)=(Ofo/OZ)_l.

Ot

Conversely, if v e C l ( [ 0 , ao), 7-/(G1)NC(Gz)) solves (13) and (14) with

v(z, t)r

in G1 x [0, a0), then

1/v

belongs to

Cl([O, ao),Tt(G1)nC(G1))

and f belongs to CI([0, ao),

7-I(G1)NCI(Gt)),

where

Ozf(Z,

t ) r Hence, f is locally univalent. The definition of f implies f(0, t ) = 0 for tE [0, a0). Furthermore,

fo z dp fo z Ofo

f(z, 0) -- v(p, 0) -

~ d p = f o ( z ) - f o ( O ) = f o ( z ) .

Thus the conditions (2) and (3) are fulfilled.

A simplified proof for a moving boundary problem for Hele-Shaw flows in the plane 105

If v solves (13), then the same reasoning as above gives

Of

^{- i}

For tC (0, a0) the term in the brackets is holomorphic in G1, hence,

Of ( ( O f ) -1)

Of h _~z Tt = k ( t ) , ot

a constant depending on t. Inserting z = 0 , this shows that k(t)--0, hence (12) is satisfied.

Finally from the holomorphy of

(l/h)(Of/Ot)(Of/Oz) -1

we obtain (8) and (11).

Remark

1. An analogous statement is valid for f C C 1 ([0, a0), 7-/(G1) M C 1 '~ (G1)) and vECI([0, a0),

7-I(G1)MC~(G1))

instead of f E C I ( [ 0 , a0),

~-~(G1)NCI(G1)) and

v E C~([0, a0), ~t(G~)NC(G1)).

The lemma of equivalence just proved makes it possible to restrict ourselves to the problem (13) and (14). This is a scale-type problem. Thus it remains to show how we can interpret the problem (13) and (14) as a special case of (4) (see Section 3).

There is a gap between Richardson's mathematical model and Lemma 1. In Lemma 1 we obtain in the converse direction merely the local univalence of

f(z, t).

But the following statement holds:

Suppose, that

(i) the initial value

fo(z)

from (2) is an analytic and univalent function in G r , r > l ;

(ii) the family {ft (z)} of analytic functions belongs to C([0,

T], ?-l(Gr,) M C(G~,)), r l ~ r .

Then there exists a positive constant

To(r')

such that

ft(z)

is univalent in Gr, for all

tE[O, To(r')).

Using this statement the conditions

(i) univalence of the analytic function

fo(z)

in G~;

(ii) vECl([0,

a0), 7-/(G~,)MC(G~,)) with

v(z,

^{t ) r}

imply the univalence of

f(z, t)

for small t in a neighbourhood of {Izl < 1}.

In Chapter 3 we shall prove the existence of such functions

v=v(z, t)

^{as solu-}

tions of a modified problem to (13) and (14).

2. A b o u t t h e a c t i o n o f a n o p e r a t o r Tt r e p r e s e n t i n g a c o n t i n u a t i o n o f Tt in s o m e B a n a c h s p a c e s

Let v be in C([0, T],

H(Gr)MC(Gr))

with r > l . Then

Tt(v)

belongs to 7-/(G1)

106 Michael Reissig and Lothar v. Wolfersdorf

for each t 6 [0, T]. But moreover

Tt(v)

possesseS an analytic continuation in a larger domain depending on G~ and

Kb

from (9).

L e m m a 2.

For an arbitrary vC~(Gr)QC(Gr) the image Tt(v) of the non- linear operator Tt applied to v can be analytically continued into

Gro

with

r0--min(b, r).

Proof.

From (15) we get

T (v) = G1

-1/o

_{27ri G1}

Iv(Q)I:g(Q '

dQQ

v(Q)v(1/0)g(Q,

1/Q,t) Q+z dQ

Q--z Q for all z E G1.

The assumption

vC~(Gr)MC(Gr)

and (9) guarantee that the kernel of the integral is holomorphic in the set

{1/ro<iQi<ro}\{z}

for all te[0, T] and

z6G1.

Conse- quently,

Tt(v) = ~ 1 / o c a v(Q)v(1/~)g(Q, 1/Q,t) Q-zQ+Z dpQ

for all

z6G1

and l < a < r 0 . Obviously, the right-hand-side can be defined for all

z 6 Ga,

and

Tt (v)

possesses an analytic continuation

1 /o v(Q)v(1/~) g(Q,

l/Q, t) Q+z dQ

(16)

Tt(v)= ~ ca Q-z Q

belonging to

?-t(Ga).

Since

Gro=Ul<a<to Ga

the operator Tt maps ~ ( G r ) into

~ ( G r o ) . For all

z6G1

we conclude

Tt(v)(z)=Tt(v)(z).

Hence

Tt(v)

represents an analytic continuation of

Tt(v)

for

v67-l(Cr)MC(Gr)

into Cr o.

There arises the question whether it is possible to estimate the action of Tt as a mapping of a Banach space B into itself. In the next lemma we shall give a positive answer for the c a s e

B=7~(Gp)NC(Gp),

l < p < r 0 .

L e m m a 3. (a)

For every function v from 7-l(Gp)MC(Gp) the following esti- mate connecting the norms Iivllp=SUPap

Iv]

and ]]Tt(v)iip=supa p

]Tt(v)]

holds:

I[T (v)ll

C(p,g)llvll p,

where the constant C is independent of vCT-l(Gp)•C(Gp) and

tE[0, T].

Moreover, we obtain for all vl, v2 e B with Iivl ilp, ][v2iip < R the Lipschitz condition

]ITt(Vl ) - ~(v2 )lip ~ 2C(p, B)RIIvl - v2iip.

A simplified proof for a moving b o u n d a r y problem for Hele-Shaw flows in the plane 107

(b)

The family of operators {Tt(v)}tc[O,T] depends continuously on

tC[0, T].

This means

lim []Ttl (v)-Tt2 (v)[[p = 0

for all v 9 7-~(Gp)NC(Gp).

tl---+t2

Proof.

(a) Let us remember that

1 ~o v(o)v(1/O)g(Q, 1/Q,t) O+z do

Using the holomorphy of

v(o)v(1/~)g(o, 1/6, t)(o+z)/o

in { l / p < IQI <P}, we obtain for all

z E OGp,, p~--+p,

and t E [0, T]

1 fo v(O)v(1/O)g(o, 1/o,t) Oo__~ z do

= G l j , o

+ 2-~il f ouo(z) v(o)v(1/O)g(o, 1/o,t)~_ dO0 '

where

Lta(z)

is a sufficiently small neighbourhood of z contained in G v. From Cauchy's integral formula and a simple estimation it follows that

~2~ 1 9 . d~o

Tt(v)(z>]<_

^{1 ] 0}

V(pC~)v(pe'~)g ,l~[pe ~,pe-i~, t) ei~/P+ ze,~/p_~:_~

1/z,t) l

< IIvll 2 sup

g(z, 1/z,t) (2~ Izl+l/p'~

- (z,t)e{1/p<~

<pI•

[zl--1/P]

for all

z E OGp,.

But the continuity of v in Gp guarantees t h a t the last inequality remains valid for all

z E OGp.

Hence, by the maximum principle

IIT,(v)ll~ = sup

IT,(v)(z)l < C(p, g)llvll~

zCGp with

C(p,g)=

^sup

Ig(z, 1/z,t)l(2+P2+l~

(z,t)C{1/p<lzl<p} • ] p=~='~ l ] "

By (9) and l < p < r 0 < b the constant

C(p, g)

is finite. The same reasoning leads to the Lipschitz condition.

108 Michael l=teissig and L o t h a r v. Wolfersdorf

(b) As in the proof of (a) one deduces / p 2 + l ~

IITtl(v)(z)--Tt2(v)(z)llp<~_

~ 2 ~ - p - - ~ 1 ) sup

Ig(z,1/z, tl)--g(z, 1/z,t2)] <__~

zC{1/P<lzl<V}

for ]tl--t2] sufficiently small and all l < p < r0, taking into consideration the uniform continuity of g in

{1/p<

]z] _<p} • [0, T].

Remark

2. It is possible to prove a corresponding inequality between

[]V]]p,~

and

]lTt(v)]]p,~,O<a<l,

where

I]V[[p,~

denotes the HTlder-norm of

vETI(Gp)AC~(Gp).

The proof of

]lTt(v)[]p,~ <_C(p, a, g)[]vl]2,~

is omitted.

For proving a regularity result for

(Of/Oz) -1

in the sense of the results in [3]

the next lemma represents an essential tool. For the formulation of this lemma we choose the following family {Er}r>0 of Banach spaces of entire functions:

{ E j ~ > 0 = {v 9 n ( C ) : sup [v(z)e-~lzl I = IIvll~ < ~}~>0"

z T C

Now we are choosing g = l in (16).

L e m m a 4.

The operator

fo O+ z dp

1 v(o)v(1/~) p - z p

~ ( v ) ( z ) = ~ i Go

z 9

a > l

arbitrary, maps E~ into itself, where I]T(v)]]~<~ exp(5r/2)]M] ~.

Moreover, we obtain for all Vl, v2 9 E~ with ][Vl ]]~, []v2 [1~ <R the Lipschitz con- dition

[IT,(vl) - T t (v2)lit <-

~ Re5~/2 IIvl -v2 I1~.

Proof.

Supposing v 9 E~ the above-defined function T(v)(z) makes sense for all zCC. This follows from the fact that

v(~)v(1/~)(p+z)

is holomorphic in C\{0}.

Hence T(v) is an entire function.

Now let us fix z 0 9 with [z0[>2. Then as in the proof of Lamina 3(a) we arrive at

1 fro v ( o ) v ( 1 / O ) ~ dO+2V(Zo)V(1/2o)

~(v)(z0) = ~ G~/~ Q

for an arbitrary b>[z0[, and

1 ~o v(p)v(1/~e_~/]~]e~(1/]~]_lzor ) p+zo dp

T(v)(zo)exp(-r]zo]) = ~ a~/b p-zo p

+ 2v(zo)~-~lzol~(1/~o)

A simplified proof for a moving b o u n d a r y problem for Hele-Shaw flows in the plane

But this leads immediately to

I ( ) ( ~ I

5 m a x Iv(z)l maxlv(z)e-'l~lle-'(b-I~~

'Cv"z~176 < ~

Izl=~/b Izl=~

+ 2 max Iv(z)l

Mzo)le -'lz~

1~1=1/2

< ~ max Iv(z)l Ilvll,.

lzl=l/2

if one takes into account that

Izol+l/b Izol+89

Izol-1/~ -< Izol-~ ___s for Izol__2, b>O

and

e~(b-lz~

for Iz0l~b.

From the definition of I[vl[~ we obtain

max

Iv(z)l__ Ilvll~e ~/2

and max

Mz)l <_ Ilvlbe ~r.

Izl=l/2 Izl=2

Thus it is possible to draw the following two conclusions:

1 l e t / 2 ~1 2

IT(v)(zo)e-rl~~

v ~ f o r e a c h z o e C w i t h l z o [ > 2 , and

109

I~(v)(zo)e-.,zo, L

^<

maxl@v)(z)e-2~ e2r

< 1 1 . s ~ / 2 .,

2

- - I z l = 2 . . . "3-~ ^{~ v,}

for each

zoeC

with

Iz0l<2.

But these conclusions yield liT(v)II~ < ~ e5r/2 II vii 2.

The same reasoning gives the Lipschitz condition.

In this section we introduced the operator Tt(v) and studied some of its prop- erties as for example the relation between

Tt

and Tt. The results obtained are useful in examining the problem

The restriction of a solution v e C l ( [ 0 ,

ao),7-l(G~)NC(Gr))

of this problem to

(z, t)eG1 •

[0, Co) represents a solution v e C l ( [ 0 , a0), T/(G1)f~C(G1)) of (13) and (14).

110 Michael Reissig a n d L o t h a r v. Wolfersdorf

3. T h e p r o b l e m (17) a n d (14) as a s p e c i a l c a s e o f (4)

To apply Theorem 1 to the problem (17) and (14), we only have to show that the conditions (5)-(7) are fulfilled. T h e assumptions concerning fo and h guarantee the existence of constants 1 <r2 < b and R > 0 such that

R ~ ]v0(z)l =

[(cOfo/Oz)-l[

in Gr2,

and

hEC([O,T],?i(Gr2)nC(Gr~)).

For a fixed

1<rlKr2

let us choose the Banach space scale

sup

I.I}oQ<l.

Grl+S(rm-rl)

Following L e m m a 1 (v(z, t ) # 0 ) it is necessary to choose the sphere

{w 9 Bs:

I1 11 < R}.

Introducing

w(z,t)=v(z,t)-vo(z),

this implies a homogeneous initial condi- tion. Thus the problem (17) and (14) can be transformed to

(18)

Owe9~ s = = -,W+VO,Oz,hTt,w+vo,j+hTt,w+voJoz,W+Vo,

( ~ - - l

-

( ~

~

f ~ - - f

c9

(19) w ( z , 0 ) = 0 .

L e m m a 5.

The operator

s

satisfies in the above-introduced Banach space scale {Bs,

[['[Is}0<s_<l

the conditions

(5)-(7)

of Theorem 1.

Proof.

Every space Bs forms a Banach algebra. Consequently, from Lem- ma 3(a),

voeB1

and

heC([O,T],B~)

we conclude that

h:Ft(W+Vo)eB~

for all 0 <

s < l and all

wEBs.

Using the result of Tutschke [8] that

O/Oz

is a bounded opera- tor as the mapping of B~ into B ' s with

][O/Ozlls__. s, <_ ((r2-rl)(S-S')) -1

one obtains

s

for every

(t,w)6[O,T]x{w6B~:[[w[[~<R}.

From L e m m a 3(b) it fol- lows that for a given

wEBs

the term

Tt(w+vo)

depends continuously on t. But this leads to limtl-~t2 [[s w ) - s w)[[s, = 0 for all tl, t2 E [0, T] and all

w6B~.

This proves (5).

Let us further consider the difference

o 0

s wl)-Co(t, w2) = - ( w l - w 2 ) (h~(wl+Vo))-(w2+vo)-~z (h(Tt(wl+vo)

_~t(w2..~Vo)))~_h(~t(Wl..~Vo)_~t(w 2 +v0)) (wl +v0) 0

A simplified proof for a moving b o u n d a r y problem for Hele-Shaw flows in the plane 111

Using

O ( w l - w 2 ) <__

2C(p,g)(R+llvolll)llwl-w211p

p

for all

Wl,W2ET-t(Gp)NC(Gp)

with IlWlllp,

Ilw2llp<R

and all te[0, T] the following estimates are valid in {Bs, I1' L}0<8<1:

IIT~(wl+v0)L ]lz;0(t, wl)-s w2)l[~, _< IlWl-W211~ Ilhlh ( ~ )

Ilhlll llw2 + volls ii~t (wl + vo ) _ ~t(w2 + vo ) ]l~

(s-s')(r2-r~)

+llhl[lllTe(Wl+Vo)-T~(w2+vo)L (s-~')(r~-~l) Ilw~+v0L IlWl-W21J~

+ Ilhl] 1 I[Tt (w2 +vo)I[~ (r2

- r l ) ( s - s')

(s - s')(r2-rl)LlhllI(R+ IlVolll )26C(r2, r~, g)

with

r ~ + l

C(r2, r~, 9) =

sup

Ig(z, 1/z,t)] (2 +r--~_ l ) .

(z,t)E{1/r2<l~l<r2 } x [O,T]

So, also (7) is proved.

Finally, in the same manner it can be verified that

with a certain constant K independent of s and t. Hence also (6) is true, which completes the proof of this lemma.

Now the application of Theorem 1 to the problem (18) and (19) yields one and only one solution

w e C 1 ([0, ao(1 - s ) ) , ~(G~+~(~2_~)) eC(G~+~(~2_~)))o<8<l with suPo~,+~(.2_.~ )

Iw(z, t)l <R

for all t e [0, a 0 ( 1 - s ) ) .

But then

v(z, t)=w(z, t)+vo(z)

represents a solution

"0 9 C 1 ([0, do(1 - s ) ) , 7-t(a~, +~(~_~,)) A c(O~+~(~,_~,)))0<~<l

of the problem (17) and (14) with supa~l+.(~2_~)Iv(z, t)] >0 for all t 9 [0, d o ( i - s ) ) . The coincidence of the operators

Tt

and

Tt

for all

v 9

guarantees that the restriction of

v(z,t)

to CI([0, ao),7-t(G~)NC(G~,)) is a solution of (13) and (14) with supa.,

Iv(z,t)[>O

for all t 9 ao). From this result together with Lemma 1, the end of Chapter 1 and the equivalence of (12) with (8) and (11) we get the following theorem concerning problem (8), (2) and (3).

112 Michael Reissig and L o t h a r v. Wolfersdorf

T h e o r e m 2. Suppose that

(i) the real-valued function g--g(z,~,t) is continuous in {Izl--1}• T] and possesses a holomorphic extension into a circular ring Kb={1/b<lzl<b} for all te[O,T];

(ii) the function h=h(z,t) belongs to the space C([O,T],?-I(Gr2)NC(Gr2)), l<r2<b, Gr2={Izl<r2}, where h(0, t)=0, h~(O,t)~O and h(z,t)~O for all (z,t)e {0 < Izl < 1} x [0, T];

(iii) the function fo(z), f0(0)--0, is holomorphic and univalent in Gr2.

Then for every l <rl <r2 there exist a positive constant ao(rl) and a uniquely determined function f = f ( z, t ), holomorphic and univalent in G~I , belonging to C 1 ([0, ao(rl)), 7-/(Grl )

n C 1 (erz))

and satisfying the conditions

R e ( h ( ~ , t ) ~ , l Of ( z , t ) ~ z ( Z , t ) ) = g ( z , 2 , t ) for all (z,t) E{Izl=l}x(O, ao(rl));

( 1 0 f ( z , t ) ~ z ( Z , t ) ) ( O , t ) _ _ O fortE(O, ao(rl));

Im h(z, t--) Ot

f(z,O)=fo(z) for zCd~l;

f(0, t ) = 0 for te[O, ao(rl)).

As a conclusion from Theorem 2 we immediately get a statement concerning the classical Hale-Shaw problem in the plane (h(z, t)=z, g(z, 2, t ) z l ) .

Corollary 1. Under the assumption that the function fo(z), f0(0)=0, is holo- morphic and univalent in G~2, for every l < r l < r 2 there exist a positive constant ao(rl) and one and only one holomorphic and univalent in Grl function f - - f (z, t)eC1 ([0, ao(rl)), 7-/(Grl )NCI(Grl )) satisfying

R e ( l ~ t ( z , t ) ~ z ( Z , t ) ) = l for (z,t) e ( l z I~-l}•

t ) ~ z ( z , t ) } = 0 for te(O, ao(r~));

f(z,O)=fo(z) for z e G ~ ; f ( 0 , t ) = 0 for tE[O, ao(rl)).

Remark 3. In connection with the moment problem for holomorphic functions Gustafson [3] studied the conditions

t. Of rz, t cosn~----(zn+bn)/2

R e \ z Ot" )-~z ~ ) = s i n n ~ _ ( z n _ Z n ) / ( 2 i ) on [zJ=l

A simplified proof for a moving boundary problem for Hele-Shaw flows in the plane 113 instead of (1).

T h e s e conditions are special cases of (8), (9) a n d (10). T h e conditions (8)-(10) represent t h e m o s t general conditions for a successful a p p l i c a t i o n of t h e non-linear a b s t r a c t C a u c h y - K o v a l e v s k y T h e o r e m due to Nishida [5].

Remark

4. C o m p a r i n g (11)

(h(z,

t ) - - z ) w i t h G u s t a f s s o n ' s condition fz(0, t) > 0, it is easy to see t h a t this a s s u m p t i o n leads to (11). Hence the solutions of T h e o - r e m 2 for t h e classical Hele-Shaw p r o b l e m coincide w i t h t h e solutions c o n s t r u c t e d b y G u s t a f s s o n in [3]. O n the o t h e r h a n d , since

h(z,t)~hz(O,t)z

as z - * 0 , (11) is equivalent to the r e p r e s e n t a t i o n fz(0, t ) = e x p ( i c ~ ) e x p ( g ( t ) ) if we a d d i t i o n a l l y sup- pose t h a t hz(0, t) is real-valued (c~ is a real c o n s t a n t ,

g=g(t)

a real-valued continuous function). T h u s , (11) really generalizes t h e condition

fz(O,

t ) > 0 .

Remark

5. F r o m T h e o r e m 1 applied to p r o b l e m (18) a n d (19) one o b t a i n s the e s t i m a t e s u p ~ l

I(Of(z, t)/Oz)-11< II (Ofo/c3z) -1 lit2 +R,

where

f = f(z, t)

is the solution f r o m T h e o r e m 2 a n d R fulfills

II(Ofo/Oz)-lllr2 >R

for all z C G r 2 .

T a k i n g a c c o u n t of R e m a r k s 1 a n d 2 a n d t h e result of [8] t h a t t h e o p e r a t o r

O/Oz

is b o u n d e d as a m a p p i n g of

7-I(Gp)NC~(Gp)

into

7-I(Gp,)NC~(Gp,); (p'<p,

0 < c ~ < l a n d

IlO/OZllp_~ p, <C/(p-p')),

we are able to prove a result c o r r e s p o n d i n g to T h e o r e m 2 b a s e d on t h e scale of B a n a c h spaces

{S~, I1" I1.}o<~_1 -- {~ (C,., +~(,.~_,.~)) nC '~ (Grl nt_8(I..2_r1)), I1" I1~,'~ }"

For in general a smaller interval t E [0, b0) a n u p p e r b o u n d for the H h l d e r - n o r m of

(Of(z, t)/Oz) -1

in G~I c a n b e o b t a i n e d b y ]1

(Ofo/Oz(z))-i I1~,~ +R

w i t h t h e s a m e R as in t h e case of t h e s u p r e m u m - n o r m s .

4. A b o u t t h e c o i n c i d e n c e o f t h e s t r u c t u r e s o f

(Ofo/OZ) -1

a n d ( O f ( z ,

t)/cgz) -1

G u s t a f s s o n p r o v e d in [3] t h a t , if t h e initial value

fo(z)

is a univalent p o l y n o m i a l or a univalent r a t i o n a l function in a n e i g h b o u r h o o d of Izl < 1, t h e n the solution of (1)-(3) is as a function of z of the s a m e s t r u c t u r e as

fo(z),

which m e a n s a univalent p o l y n o m i a l or a univalent r a t i o n a l function. I n t h e p o l y n o m i a l case this coincidence of the s t r u c t u r e s can be expressed b y the aid of t h e derivatives in t h e following form:

I f

Ofo/Oz

is a p o l y n o m i a l which has no zeros in a n e i g h b o u r h o o d of Izl < 1 t h e n also

Of(z, t)/Oz

is a p o l y n o m i a l which has no zeros in a n e i g h b o u r - h o o d of ]z I < 1 for t f r o m a suitable right-sided n e i g h b o u r h o o d of t = 0.

Such a f o r m u l a t i o n c a n n o t b e d e d u c e d for t h e r a t i o n a l case f r o m t h e results of [3].

114 Michael Reissig and L o t h a r v. Wolfersdorf

Using (Ofo/OZ) -1 and (Of(z, t)/Oz) -1 the last statement concerning the deriva- tives Ofo/OZ and Of(z, t)/Oz gets a new formulation.

If (Ofo/OZ)-l= 1/P(z), where P(z) is a polynomial without zeros in a neighbourhood of [z[ <_ 1, then (Of(z, t)/Oz) -1 = 1/Q(z, t), where Q(z, t) is a polynomial in z without zeros in a neighbourhood of [z[ _< 1 for every t from a right-sided neighbourhood of t = 0.

In the following we are interested in the proof of a result of the same type. For this purpose, let us choose with arbitrary 0 < Sl < s2 the Banach space scale of entire functions

where the spaces E r were introduced in Section 2.

T h e o r e m 3. In addition to the assumptions of Corollary 1 suppose that (Ofo/Oz) -1 is an entire function belonging to E~ 1 . Then it is known that be- sides the statement of Corollary 1, there holds (Of(z, t)/Oz) -1 ECI([0, do(S2)), Bs2) with s2> sl and a certain positive constant do(S2). In particular this means that (Of(z, t)/Oz) -1 is an entire function for all t e [0, do(S2)).

(In [0, do(r1, s2)), do(r1, s2)=min(ao(s2), do(r1)), b o t h properties of f ( z , t) are fulfilled.)

Proof. It remains to prove the statement for (Of(z,t)/Oz) -1, which follows from the application of T h e o r e m 1 to the problem (18) and (19), equivalently, (17) and (14) with the scale {Bs, [[. [[s}0<:s_<l.

Prom Lemma 4 the continuity of Tt(v) as a mapping of B~ into itself is clear.

Hence we only have to study the behaviour of the differential operator O/Oz and the multiplication operator z. in the scale {Bs,

I1"

L}0<~_<l.

For the first let v be a function from Er. Applying Cauchy's integral formula in a small neighbourhood Ua(zo) of a fixed point z0 we obtain

Ov rzo 1 .~2~ v(o)e-~lOle~(iol-lzol)

= - - d~

~ ( z 0 ) e - 2~" - v 0 - z o

with 0 = z0 + a exp(i~). Using I I 0 l - [ z o

I I

-<[o- zo [ this relation leads to

~

(z0)e-~l*ol <

Ilvll~e~

With a = l / r we have [lOv/Ozllr<_erlfvll~. Hence O/Oz is a bounded operator from E~, respectively, from B~ into itself. In the second place let v be a function from

A simplified proof for a moving boundary problem for Hele-Shaw flows in the plane 115

/{Jr. Then it cannot be expected, that zv belongs to/{Jr. For example, let us choose v=exp(2z) EE2. Then

supIze 2ze-21z][ _> sup [ x e x p ( 2 x ) e x p ( - 2 x ) [ - - o o

z E C x c R +

as x tends to infinity. But if we consider z as a mapping of Er into Er, with rt>r, then

, 1

[Izvlb = z~cSUp [zve-~'~z~ ] < z~cSup rw-~Jzl ] supz~c Izfe-(~ -~)Izl _< IIvfl~ e ( r ' - r---)"

Hence the multiplication operator z. is a bounded operator in the scale {Bs,

I1 IJs}

with [[zvl[s, _< [[vlls/(e(s2 -81)(8-8t)).

As in Lemma 5 one proves that the oprator s from (18) satisfies the conditions (5)-(7) from Theorem 1. The application of this Theorem to (18) and (19) with the scale {Bs, II"

L}0<~_<l

yields the statement for (Of(z, t)/Oz) -1. This completes the proof.

Remark 6. Using the scale {Bs, I1 Ilj0<~<l one can also get the univalence of f ( z , t ) from that of fo(z). Let us suppose ](Ofo/Oz)-l]>_R>O in Gr2 and fix the sphere Ilv-(Ofo/OZ) -1 ]l~ < R e x p ( - r 2 s 2 ) around (Ofo/Oz) -1. Then the application of Theorem 1 to the problem (18) and (19) leads to

I[v(z, t) - (Ofo(z)/Oz) -1 ]Is = I[ (Of(z, t)/Oz) -1 - (Ofo(z)/Oz) -1 I[~ < Re-r2s2"

But this means that

max

I(Of(z, t)/Oz) - 1 - (Ofo(z)/Oz) -1

le -(sl+(s2-s~)(1-s)lzb) < Re -~2~2,

Gr 2

and

m_ax ](Of(z, t)/Oz) -1 - (Ofo(z)/Oz)-ll < R.

Gr 2

Hence O f ( z , t ) / O z r for all z E G ~ and all suitable te[0, ao(8:)). T h e n an upper bound for

ll(OY(z, t)/Oz) -111~ is II(Ofo(zD/Oz) -111~x + R e - r ' = .

But we point out that the restriction to the above-introduced sphere around (Ofo/Oz) -1 can reduce the interval of existence of the solution with regard to t from Corollary 1.

Note. The authors thank the referee for the information about a new reference which gives more of the history and the physical background for equations (1) till (3) and which also contains an up-to-date bibliography for it: S. D. Howison:

Complex variable methods in Hele-Shaw moving boundary problems, preprint 1991 (Mathematical Institute, Oxford OX1 3LB, United Kingdom).

116 Michael Reissig and Lothar v. Wolfersdorf:

A simplified proof for a moving boundary problem for Hele-Shaw flows in the plane R e f e r e n c e s

1. ATHANASSOULIS, g . A. and MAKRAKIS, g . N., A function-theoretic approach to a two-dimensional transient wave-body interaction problem, W M A G Techni- cal Report 3-89, National Technical University of Athens, 106.82 Athens 42, Greece.

2. DUCHON, J. and ROBERT, R., Estimation d'op@rateurs int@graux du type de Cauchy dans les ~chelles d'Ovsjannikov et application, Ann. Inst. Fourier (Grenoble) 36 (1986), 83-95.

3. GUSTAFSSON, B., On a differential equation arising in a Hele-Shaw ftow moving bound- ary problem, Ark. Mat. 22 (1984), 251-268.

4. KANO, T. and NISHIDA, T., A m a t h e m a t i c a l justification for Korteweg-de Vries equa- tion and Boussinesq equation of water surface waves, Osaka J. Math. 23 (1986), 389-413.

5. NISttIDA, T., A note on a theorem of Nirenberg, J. Diff. Geom. 12 (1977), 629-633.

6. OVSJANNIKOV, L. V., The Cauchy problem in a Banach space scale of analytic func- tions, continuum mechanics and related problems of analysis, in Conference 23-29.9.1971 in Tbilisi (Georgia), vol. 11, pp. 219-229, 1974. (Russian) 7. RICHARDSON, S., Hele-Shaw flows with a free b o u n d a r y produced by the injection of

fluid into a narrow channel, J. Fluid Mech. 56 (1972), 609-618.

8. TUTSCHKE, W., Solution of Initial Value Problems in Classes of Generalized Analytic Functions, Teubner, Leipzig, and Springer-Verlag, Berlin Heidelberg, 1989.

9. VINOGRADOV, Yu. P. and KUFAREV, P. P., About a filtration problem, Prikl. Mat.

Mekh. 12 (1948), 181-198. (Russian)

Received June 4, 1991 Michael Reissig

Department of Mathematics Bergakademie Freiberg Bernhard-von-Cotta-Str. 2 0-9200 Freiberg

Germany

Lothar v. Wolfersdorf Department of Mathematics Bergakademie Freiberg Bernhard-von-Cott a-Str. 2 0-9200 Freiberg

Germany